Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1. Hidden Figures – another maths movie

Last week we looked at the film “Good Will Hunting” and explored one of the
mathematical problems that appears in it. I also posed a question about another
film: “What is the title of the film that celebrates the role of the pioneering
women mathematicians at NASA in the 1960s?”

Most of you correctly identified the film as Hidden Figures. Let’s start by
watching the trailer.

1 mark

1.1 Just to check that you were concentrating on that trailer, for how many
hours did the children say their mother had been gone?

3 hours

30 hours

300 hours

3,000 hours

30,000 hours

The answer is 300 hours, which the mother immediately converts into 12½ days.

The film tells the story of three brilliant black women who helped send the
American astronauts into space. These are just a few of the many women who
played a crucial role in the space race. It is a terrific film, which received
three Oscar nominations and which is both inspiring and exciting. Well worth
watching if your parents or teacher can get hold of the DVD.

2. Moore's Law

The women were known as “computers”, because the original term computer
referred to a person who did calculations. Only later did the term computer
become more associated with machines. Increasingly, machine computers at NASA
took over from human computers, and many of the women became some of the first
computer programmers and operators. Those early computers were incredibly
difficult to operate and relatively slow, certainly compared to today’s
computers, as discussed in a blogpost on the ZME Science website:

…back on Earth at the Goddard Space Flight Center thousands of flight
technicians and computer experts employed the IBM System/360 Model 75s
mainframe computer to make independent computations and maintain communication
between Earth and lunar landers. These computers cost $3.5 million a piece and
were the size of a car. Each could perform several hundred thousand addition
operations per second, and their total memory capacity was in the megabyte
range. Programs were developed for the 75s that monitored the spacecraft’s
environmental data and astronauts’ health, which were at the time the most
complex software ever developed. Today, however, even a simple USB stick or
WiFi router is more powerful, let alone an iPhone. The iPhone 6 uses an
Apple-designed 64 bit Cortex A8 ARM architecture composed of approximately 1.6
billion transistors. It operates at 1.4 GHZ and can process instructions at a
rate of approximately 1.2 instructions every cycle in each of its 2 cores.
That’s 3.36 billion instructions per second. Put simply, the iPhone 6’s clock
is 32,600 times faster than the best Apollo era computers and could perform
instructions 120,000,000 times faster. You wouldn’t be wrong in saying an
iPhone could be used to guide 120,000,000 Apollo era spacecraft to the moon,
all at the same time.

Transistors are important components of computer circuitry, and the more transistors a computer circuit has, the more calculations it can do at one time.

This remarkable improvement in computing is known as Moore’s Law. Wikipedia
says:

Moore's law is the observation that the number of transistors in a dense
integrated circuit doubles approximately every two years. The observation is
named after Gordon Moore, the co-founder of Fairchild Semiconductor and Intel,
whose 1965 paper described a doubling every year in the number of components
per integrated circuit, and projected this rate of growth would continue for
at least another decade. In 1975, looking forward to the next decade, he
revised the forecast to doubling every two years

You might not find this doubling of the number of transistors on a computer
circuit particularly impressive, but it is actually mind-blowing. It has
literally changed civilisation.

When something doubles, and doubles again, and again and again … then the
results are phenomenal. It is a type of increase known as exponential growth.
The graph below shows Moore’s Law between 1971 and 2011. Take note of the funny
scale on the y-axis, which increases by a factor of 10 at each step. So, in 40
years, the number of transistors on a microprocessor increased from 2,300 to
2,600,000,000, which is an increase by a factor of more than a million.

1 mark

2.1 Look at the number of transistors on the 80486 microprocessor (~1990)
and the number of transistors on the AMD K8 microprocessor (~2003). By what
factor did the number of transistors increase in just 12 years?

2

10

50

100

500

The 80486 had roughly 1,000,00 transistors, and the AMD K8 had about 100,000,000
transistors, so there was an increase by a factor of 100.

3. The Chess Problem

There is a very old story that demonstrates the power of exponential growth and
Moore’s Law, but it relates to chess and rice, not computers and transistors.

In India, or Persia or somewhere east of the English Channel, the King or
Maharajah or Sultan wanted to reward a mathematician, because he had just
invented chess. The King offered a chest of gold coins or a barrel of diamonds,
but instead the mathematician asked for a grain of rice on the first square of a
chessboard, 2 grains on the second square, 4 grains on the third square and so
on, doubling the number of grains of rice on each square until the 64th square.

The King laughed at what seemed like a trivial reward, but he did not laugh for
long.

Grab a piece of paper and work out the number of grains of rice on each of the
first 32 squares. You will almost certainly need a calculator, and make sure you
check your answers. I have given you a headstart by listing the answers for the
first 10 squares.

1 mark

3.1 How many grains were on the 15th square?

Correct Solution: 16,384

If the tenth square has 512 grains (from the table), then you should have
worked out (11th = 1,024), (12th, 2,048), (13th, 4,096), (14th, 8,192) and
(15th, 16,384). The answer is 16,384.

1 mark

3.2 How many grains were on the 20th square?

Correct Solution: 524,288

The answer is 524,288.

1 mark

3.3. How many grains were on the 32nd square?

Correct Solution: 2,147,483,648

The answer is 2,147,483,648.

You may have worked out that the number of grains on the nth square is 2n–1. So, instead of doubling over and over again, you could use the xy button on your calculator.

If you carried on the calculation for the full 64 squares, then you should
arrive at 9,223,372,036,854,775,808 grains on the 64th square.

If you want to add up all the grains on all the squares, then the total is double
the number of grains on the final square minus 1. (You can check this by looking
at the table above. The total number of grains for the first, say, five squares
is 1 + 2 + 4 + 8 + 16 = 31, which is the same as [2 × 16] – 1 = 31.)

This means that the total number of grains on all 64 squares is:
(2 × 9,223,372,036,854,775,808) – 1 = 18,446,744,073,709,551,615.

That is about 100 billion tonnes of rice, which is about hundred times more rice
than the whole Earth grows each year. That is the power of exponential growth.

2 marks

3.4 In another version of the story, the King gets revenge by saying that the mathematician can only have the grains of rice if he can count them. If the mathematician counts at the rate of 1 grain per second, how long will it take him to collect his reward of 18,446,744,073,709,551,615 grains? Which of these answers is closest?

400 years

500 million years

600 billion years

700 trillion years

800 quadrillion years

To count 18,446,744,073,709,551,615 grains will take 18,446,744,073,709,551,615 seconds, which is roughly 18×1018 sec = 3×1017 mins = 5×1015 hours = 2.1×1014 days = 5.8×1011 years = 580 billion years, so the answer is roughly 600 billion years.

4. Summer salaries

You get a summer job that lasts for 6 weeks. You have two salary options:

Option A – £100/week.
Option B – a salary that doubles each week, starting at £10/week.

1 mark

4.1 How much would option A deliver in total after 6 weeks? (Note: no need to include £ in your answer, as we have done that for you!)

£ Correct Solution: 600

Obviously option A delivers 6 x £100 = £600.

1 mark

4.2 How much would option B deliver in total after 6 weeks? (Note: no need to include £ in your answer, as we have done that for you!)